What is the standard model of particle physics trying to tell us?

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I will argue that the standard model contains a rather strong hint that -- instead of being simply an ordinary continuous 4D manifold -- spacetime is actually the product of a 4D manifold and a certain discrete/finite 6D space (i.e. there are 6D discrete/finite "extra dimensions"). I will introduce this idea and the evidence for it in simple way, and then discuss various outstanding puzzles and future directions.